Certain stability problems in physics lead to linear operators with a zero energy eigenvalue or resonance.
Examples are the energy critical wave equation \(\Box u - u^5=0\)  in \({\mathbb {R}}^{1+3}\)  which admits the stationary solutions \(W_\lambda (x):=\lambda (1+\lambda ^2 |x|^2/3)^{-\frac{1}{2}}\)  for \(\lambda >0\) . Linearizing around
\(W_\lambda \)  leads to \(H=-\Delta - 5W_\lambda ^4\)  which has \(\partial _\lambda W_\lambda \)  as a
resonant mode of zero energy. Another example is the critical Yang-Mills problem in dimensions \(4+1\) .
It is therefore necessary to obtain dispersive bounds in this context as well. Note that the local decay of Section REF 
easily allows for this as the asymptotic expansions in time (as derived in [1]}, [2]} for example) isolate the contributions of the threshold singularities and identifies them as being of finite rank. In case of \(L^1\rightarrow L^\infty \)  this required some additional work, see
[3]}, [4]}, and [5]} for the case of the Schrödinger evolution. Yajima [5]} obtains explicit expressions for the term \(Bt^{-\frac{1}{2}}\)  which needs to be subtracted to obtain
the \(t^{-\frac{3}{2}}\)  decay of the bulk (explicit here means that \(B\)  can be computed from the zero energy and resonance states). The wave equation in three dimensions is analyzed in [7]}. We recall the main linear result from the latter reference.
